22 problems

Let $n$ be the spatial dimension, let $\mu>0$ and $\alpha>1$, and consider the small-data Cauchy problem … with $u_1\in L^1(\mathbb{R}^n)$. Let $\gamma$ be the function determined…

For the $2$-D semilinear wave equation with scale-invariant damping … where $t\ge 1$, $\mu>0$, and $p>1$, a global small-data weak solution is a weak solution $u$ arising from suff…

Let $n\geq 4$, $\mu>0$, and consider the regular Cauchy problems … with small initial data, or … with small initial data. Denote by $p_S$ the Strauss exponent and by $p_F$ the Fuji…

Let $\Omega$ be a bounded interval, and consider the damped wave equation … where $S_0$ is the Dirichlet realization of $-\Delta+b$, $\alpha\geq 0$ is a parameter, and $a$ is the d…

Let $\mathcal{B}=\{x\in\mathbb{R}^n:|x|\leq 1\}$ be the closed unit ball and consider the exterior problem on $\mathcal{B}^c$: … with $u=0$ on $\partial\mathcal{B}^c$ and initial d…

Uniform stabilization conjecture. Equation is uniformly stable if and only if $b$ satisfies the Geometric Control Condition and the Turning Point Condition.

Let $z=z(t)$ denote the center position of a solitary wave in a global solution of the damped nonlinear Klein–Gordon equation with delta potential, and let $\sigma\in\{0,1\}$ and…

Consider wave models whose derivative-type nonlinearity has the form $|u_t|^{p_{\mathrm{crit}}(n)}\mu(|u_t|)$, where $p_{\mathrm{crit}}(n)$ is a possible critical exponent and…

Let $n\in\mathbb{N}$, let $a(x)=x^{2n}$, and consider the quadratic operator function … with level curves $\lambda_b=-c(b)+ib$, where $b\to+\infty$, and fix $\varepsilon>0$ by requ…

Let $n\geqslant 1$, let $\gamma>\tilde{\gamma}=\frac{\sqrt{n^2+16n}}{4}-\frac{n}{4}$, and consider solutions to the corresponding linear classical damped wave equation with vanishi…

Consider the one-dimensional damped wave equation with damping localized in a subset of the spatial domain, and its associated solution semigroups on the corresponding $L^1$ and…

Sharp lifespan conjecture. The sharp lifespan estimates above should still hold for every spatial dimension $n\geqslant 1$.

Consider the damped wave equation with mass parameter $u$ and damping parameter $mu$, and let $delta$ be the parameter defined in the paper by … In particular, the condition…

Let $\tau\in(0,\beta)$, and consider the linear Moore–Gibson–Thompson equation in the dissipative case together with the linear viscoelastic damped wave equation … with $\beta>0$.…

Transition-set optimality conjecture. The estimates in the two preceding theorems should be optimal except on these transition sets. In the critical case, the lifespan should satis…

Transition-curve optimality conjecture. The lifespan estimates in Theorem should be optimal except on this transition curve. On the curve,

Transition-surface logarithmic-gain conjecture. Lifespan estimates on the transition surfaces should admit a logarithmic improvement.

Takamura's conjecture. The lifespan satisfies

Consider the linear damped wave equation … with compactly supported smooth initial data $u^0(x,0)=u_1(x)$ and $u^0_t(x,0)=u_2(x)$ on $\mathbb{R}^n$. The damping is called non-effec…

Uniform asymptotic Strichartz estimate. At least in the autonomous case, the displayed estimate should hold.

Critical-exponent conjecture. The exponent $p_1$ will be the critical exponent of this damped wave equation with nonlinear memory. It is the critical exponent for the corresponding…

Consider the Cauchy problem … for the semilinear damped wave equation. Diffuse-structure conjecture. The damped wave equation should have the diffuse structure as…